While formalized programs and law schools existed during the time, many people became lawyers by "reading law." They spent an amount of time basically as an apprentice to an experienced lawyer. Lincoln apprenticed with John T. Stuart and was admitted to the bar after three years.

You might be interested to know that this process still exists in modified forms today in some states.

Sources: http://hosted.law.wisc.edu/wordpress/wilj/files/2012/02/katcher.pdf

A. Lincoln by Ronald C. White

http://mobile.nytimes.com/2014/08/03/education/edlife/how-to-learn-the-law-without-law-school.html